cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A340647 G.f.: Sum_{k>=0} x^(k^2) / Product_{j=1..k} (1 - x^(2*j))^2.

Original entry on oeis.org

1, 1, 0, 2, 1, 3, 2, 4, 5, 6, 8, 8, 14, 12, 20, 18, 31, 27, 42, 40, 60, 60, 80, 86, 111, 124, 146, 174, 199, 241, 262, 328, 353, 444, 464, 590, 620, 780, 812, 1020, 1075, 1326, 1400, 1710, 1833, 2198, 2370, 2804, 3072, 3570, 3936, 4522, 5048, 5713, 6414, 7190
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 14 2021

Keywords

Links

Crossrefs

Cf. A000700, A006950, A340623.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[x^(k^2)/Product[(1-x^(2*j))^2, {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

Formula

a(n) = A006950(n) - A340623(n).
a(n) ~ exp(Pi*sqrt(n/2)) / (4*sqrt(2)*n).

A340621 The number of partitions of n without repeated odd parts having more odd parts than even parts.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 2, 1, 4, 2, 6, 3, 9, 6, 12, 10, 17, 17, 22, 26, 30, 40, 40, 57, 55, 82, 74, 112, 103, 153, 140, 203, 193, 270, 262, 351, 357, 458, 478, 589, 641, 760, 846, 971, 1114, 1244, 1450, 1582, 1880, 2018, 2412, 2558, 3086, 3247, 3914, 4102, 4949
Offset: 0

Views

Author

Jeremy Lovejoy, Jan 13 2021

Keywords

Examples

			a(8) = 4 counts the partitions [7,1], [5,3], [5,2,1], and [4,3,1].

Links

Crossrefs

Cf. A006950, A143184, A340622, A340623.

Programs

  • Maple
    b:= proc(n, i, c) option remember; `if`(n=0,
      `if`(c>0, 1, 0), `if`(i<1, 0, (t-> add(b(n-i*j, i-1, c+j*
      `if`(t, 1, -1)), j=0..min(n/i, `if`(t, 1, n))))(i::odd)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 13 2021
  • Mathematica
    b[n_, i_, c_] := b[n, i, c] = If[n == 0,
     If[c > 0, 1, 0], If[i < 1, 0, Function[t, Sum[b[n - i*j, i - 1, c + j*
     If[t, 1, -1]], {j, 0, Min[n/i, If[t, 1, n]]}]][OddQ[i]]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 29 2022, after Alois P. Heinz *)
  • PARI
    my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=1, sqrt(N), x^(k^2)*(1-x^(2*k))/prod(j=1, k, (1-x^(2*j))^2)))) \\ Seiichi Manyama, Jan 14 2021

Formula

G.f.: Sum_{n>=1} q^(n^2)*(1-q^(2*n))/Product_{k=1..n} (1-q^(2*k))^2.
a(n) ~ exp(Pi*sqrt(2*n/5)) / (2^(3/2) * 5^(3/4) * n). - Vaclav Kotesovec, Jan 14 2021

A340622 The number of partitions of n without repeated odd parts having an equal number of odd parts and even parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 0, 3, 1, 4, 2, 5, 5, 6, 8, 8, 14, 10, 20, 14, 30, 20, 40, 29, 56, 42, 72, 62, 96, 88, 122, 125, 160, 174, 202, 239, 263, 322, 334, 431, 434, 566, 554, 739, 719, 954, 920, 1222, 1192, 1552, 1524, 1964, 1962, 2466, 2500, 3088, 3196, 3848, 4046
Offset: 0

Views

Author

Jeremy Lovejoy, Jan 13 2021

Keywords

Examples

			a(9) = 4 counts the partitions [8,1], [7,2], [6,3], and [5,4].

Links

Crossrefs

Cf. A006950, A340621, A340623.

Programs

  • Maple
    b:= proc(n, i, c) option remember; `if`(n=0,
      `if`(c=0, 1, 0), `if`(i<1, 0, (t-> add(b(n-i*j, i-1, c+j*
      `if`(t, 1, -1)), j=0..min(n/i, `if`(t, 1, n))))(i::odd)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 13 2021
  • Mathematica
    b[n_, i_, c_] := b[n, i, c] = If[n == 0,
     If[c == 0, 1, 0], If[i < 1, 0, Function[t, Sum[b[n-i*j, i-1, c + j*
     If[t, 1, -1]], {j, 0, Min[n/i, If[t, 1, n]]}]][OddQ[i]]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 29 2022, after Alois P. Heinz *)

Formula

G.f.: Sum_{n>=1} q^(n^2+2*n)/Product_{k=1..n} (1-q^(2*k))^2.
a(n) ~ exp(Pi*sqrt(2*n/5)) / (2^(1/2) * 5^(3/4) * (1 + sqrt(5)) * n). - Vaclav Kotesovec, Jan 14 2021
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.